Elasticity is a measure of responsiveness. It tells how much one thing changes when you change something else that affects it.
For example, the elasticity of demand tells us how much the quantity demanded changes when the price changes. The elasticity of demand measures the responsiveness of quantity demanded to changes in the price charged.
The following discussion mostly uses the elasticity of demand for its examples. The elasticity concept can be used for other things, too, like supply or income.